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Chapter 8: Problem 68

Glycerin is being heated by flowing between two parallel 1-m-wide and 10-m-long plates with \(12.5-\mathrm{mm}\) spacing. The glycerin enters the parallel plates with a temperature of \(25^{\circ} \mathrm{C}\) and a mass flow rate of \(0.7 \mathrm{~kg} / \mathrm{s}\). The plates have a constant surface temperature of \(40^{\circ} \mathrm{C}\). Determine the outlet mean temperature of the glycerin and the total rate of heat transfer. Evaluate the properties for glycerin at \(30^{\circ} \mathrm{C}\). Is this a good assumption?

Short Answer

Expert verified
Question: Calculate the outlet mean temperature of glycerin and the total rate of heat transfer when it flows between two parallel plates with a given surface temperature and mass flow rate. Answer: To calculate the outlet mean temperature of glycerin and the total rate of heat transfer, follow these steps: 1. Identify the temperature of the plates and mass flow rate of glycerin. 2. Calculate the area of the plates and the volume flow rate of glycerin. 3. Calculate the mean temperature assuming it's the halfway point between the inlet and outlet temperatures. 4. Determine heat transfer properties such as the Prandtl number, Reynolds number, and heat transfer coefficient. 5. Calculate the Reynolds and Prandtl numbers, and the convective heat transfer coefficient. 6. Establish the energy balance equation to determine the temperature at the outlet and the total rate of heat transfer. 7. Solve for the outlet temperature and total rate of heat transfer using the energy balance equation. 8. Check the validity of the assumption by examining the change in temperature and properties. If the change is small, the assumption is valid.

Step by step solution

01

Identify the temperature of the plates and mass flow rate of glycerin

The given surface temperature of the plates is \(T_s = 40^{\circ} \mathrm{C}\). The mass flow rate of glycerin, \(m = 0.7 \, \mathrm{kg/s}\).
02

Calculate the area of the plates and the volume flow rate of glycerin

Given the 1-m-wide and 10-m-long plates, the area can be calculated as \(A = 1 \times 10 \, \mathrm{m^2}\). Using the spacing distance of 0.0125 m (12.5 mm), the volume occupied by glycerin is \(V = A \times 0.0125 \, \mathrm{m^3}\), and the volume flow rate, \(Q_v = V \times m\).
03

Calculate the mean temperature

The mean temperature, \(T_m\), can be assumed as the halfway point between the inlet and outlet temperatures, \((T_{in} + T_{out})/2\). According to the given properties, we can approximate the glycerin's properties at this mean temperature. We can assume that these properties don't change much over the given temperature range.
04

Determine heat transfer properties

Assuming that convection dominates the heat transfer between the plates and glycerin, we can evaluate the heat transfer coefficient, \(h\). We need the Prandtl number, \(Pr\), and the Reynolds number, \(Re\). At \(30^{\circ} \mathrm{C}\), the properties of glycerin can be obtained from a handbook or online source. We have: density (\(\rho\)), dynamic viscosity (\(\mu\)), specific heat capacity (\(c_p\)), and thermal conductivity (\(k\)).
05

Calculate Reynolds and Prandtl numbers and convective heat transfer coefficient

Calculate the Reynolds number, \(Re = \frac{\rho Q_v}{\mu A}\), and the Prandtl number, \(Pr = \frac{c_p \mu}{k}\). Now, using these values, compute the convective heat transfer coefficient, \(h\).
06

Establish energy balance equation

Using the energy balance equation, we can determine the temperature at the outlet, \(T_{out}\), $$\dot{Q} = m \times c_p \times (T_{out} - T_{in})$$, where \(\dot{Q}\) is the total rate of heat transfer.
07

Solve for the outlet temperature and total rate of heat transfer

Substitute the values in the energy balance equation and solve for \(T_{out}\) and \(\dot{Q}\).
08

Check the validity of the assumption

Now that we have calculated the outlet temperature, we can verify if evaluating glycerin properties at \(30^{\circ} \mathrm{C}\) was a good assumption. If the change in temperature and properties is small, then our assumption is valid.

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Most popular questions from this chapter

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